February 17, Simplex Method Continued

Size: px
Start display at page:

Download "February 17, Simplex Method Continued"

Transcription

1 February 17, 2005 Simplex Method Continued 1

2 Today s Lecture Review of the simplex algorithm. Formalizing the approach Alternative Optimal Solutions Obtaining an initial bfs Is the simplex algorithm finite? (Answer, yes, but only if we are careful) 2

3 The simplex algorithm (for max problems) Start with a feasible corner point solution find an improved corner point solution No Is it optimal? No Yes quit with optimal solution Is the optimum unbounded from above? Yes quit with proof of unboundedness 3

4 LP Canonical Form iff there is a basis that is feasible. z x 1 x 2 x 3 x = = 6 0` = 2 The basic variables are z, x 3 and x 4. The non-basic variables are x 1 and x 2. The basic feasible solution (bfs) is z = 0, x 1 = x 2 = 0, x 3 = 6, x 4 = 2 4

5 Optimality Conditions: maximization form z 1 0 x 1 x 2 x 3 x 4 -c c = = 6 The bar indicates that it is possibly the coefficient after some pivots = 2 The bfs is optimal if all coefficients z-row are 0. Use c j to denote the cost coefficients. Opt. conditions: The bfs is optimal if 0 for all j. -c j 5

6 The Simplex Pivot Rule z x 1 x 42 x 3 x 4 x = = = = 4 Pivot in a variable whose cost-row coefficient is negative. If there are multiple choices, then use a refined rule. If 0 for all j, then the basis is optimal. -c j 6

7 Review from last lecture: To do with your partner z x 1 x 42 x 3 x 4 x = = = = 4 1. Determine the basic variables and the basic solution 2. What will be the entering variable? 3. What will be the variable that leaves the basis? Reminder: set the value of the entering variable to. 7

8 The min ratio rule z x 1 x 42 x 3 x 4s x 45 If a is 0 for all i, then the objective value is unbounded from above c -2 s 0 = 3z a2 3s 1 = b a 1s -1 a 2s 0 0 = = b b 2 Min Ratio = min{ b i / a is : a js > 0 and i = 1 to m }. Usually, the argmin value is denoted as r. In this case, r = 3. Pivot on a rs 8

9 Minimum Ratio Rule Pivot out the basic variable in row r, where r = argmin i { b i / a is : a is > 0}, and thus b r / a rs = min { b i / a is : a is > 0}. If a is 0 for all i, then the solution is unbounded. 9

10 The pivot z x 1 x 42 x 3 x 4 x = /2 0 = /2 0 = /2 1 = The objective value strictly improves so long as b r > The new (basic feasible solution) bfs is feasible. 10

11 Alternative Optima (maximization) z x 1 x 2 x 3 x 4 Recall: z + 4 x 2 = 8 Note that z does not depend on x = = = 2 This basic feasible solution is optimal! There is an alternative optimum because the non-basic variable x 1 has a 0 cost-coefficient. If x 1 enters the basis, what variable leaves? 11

12 Alternative Optima (maximization) z x 1 x 2 x 3 x c = c = 6 = 2 There may be alternative optima if -c j 0 for all j and c j = 0 for some j where x j is non-basic Use the min ratio rule to determine which variable leaves the basis. 12

13 Alternative Optima (maximization) z x 1 x 2 x 3 x = = = 28 Perform the pivot. Note: the solution is different, but the objective value is the same. 13

14 Review of notation A basic feasible solution is optimal if - c j 0 for all j. Assumption: the entering variable is x s (and so - c s < 0) Pivot out the basic variable in row r, where r = argmin i { b i / a is : a is > 0}, and thus b r / a rs = min { b i / a is : a is > 0}. If a is 0 for all i, then the solution is unbounded. 14

15 Simplex Method (Max Form) Step 0. The problem is in canonical form and b 0. Step 1. If - c 0 then stop. The solution is optimal. If we continue, then there exists some - c j < 0. Step 2. Choose any non-basic variable to pivot in with - c s < 0, e.g., - c s = min j { - c j - c j < 0 }. If a is 0 for all i, then stop; the LP is unbounded. If we continue, then there exists some a is > 0. Step 3. Pivot out the basic variable in row r, where r is chosen by the min ratio rule, that is r = argmin i ( b i / a is : a is > 0 ). Step 4. Replace the basic variable in row r with variable x s and reestablish canonical form (i.e., pivot on the coefficient a rs. ) Step 5. Go to Step 1. 15

16 Preview of what is to come Obtaining an initial canonical form Degeneracy and improving solutions Proving finiteness and optimality under no degeneracy Handling degeneracy 16

17 The Phase 1 method to obtain an initial bfs We need an initial bfs. Approach: create a new LP that is related to the original LP and has the following features: 1. It is easy to find a bfs for the new LP 2. An optimal bfs for the new LP is a bfs for the original problem. 17

18 Step 0. Start with a tableau for the original LP z x 1 x 2 x 3 x = = = 2 Note: this is different than our running example from the last lecture. 18

19 Step 1. Write the constraints of the original LP x 1 x 2 x 3 x = = 2 FACT: Once we find a basic feasible solution, we can reconsider the original cost coefficients. 19

20 Step 2. Create new variables called artificial variables which will be the basic variables for the new LP. x 1 x 2 x 3 x 4 x 5 x = = 2 We will choose an objective function soon. 20

21 Step 3. Choose an objective so that the artificial variables will be 0 in an optimal solution, assuming that there is a bfs for the original problem. w x 1 x 2 x 3 x 4 x 5 x = = = 2 The objective function is w = -x 5 x 6 max w = 0 if and only if there is a feasible solution for the original problem We use w to not confuse it z. 21

22 Step 4. Subtract constraints 1, 2 (etc) to the objective constraint so that the tableau becomes canonical w x 1 x 2 x 3 x 4 x 5 x = = = 2 22

23 Step 5. Solve the Phase 1 Problem using the simplex algorithm. BV w x 1 x 2 x 3 x 4 x 5 x 6 w = 80 x /6 12 1/6 5 1/3 1-1/2 0 = 61 x /6 1 11/6 3 2/3 0-1/2 1 = 23 x 2 23

24 Step 6. Eliminate the artificial variables, and put back in the original objective function. BV wz x 1 x 2 x 3 x 4 x 5 x 6 wz = 0 x /6 1/6 1/3-1/2 = 1 x /6 11/6 2/3-1/2 = 3 24

25 Step 7. Add multiples of constraints (1) (2), etc to get the tableau in canonical form BV -w z x 1 x 2 x 3 x 4 x 5 x 6 -w z / / = 03 x /6 1/6 1/3-1/2 = 1 x 1 x /6 11/6 2/3-1/2 = 3 Subtract 3 times constraint 1 from the objective. Add 2 times constraint 2. 25

26 Step 8. Solve the original LP starting with the previous bfs BV -w z x 1 x 2 x 3 x 4 x 5 x 6 z = x 1 0 x /3-1/2 = /3-1/2 = This tableau was reached after one more pivot 26

27 Phase 1 method: a summary 1. Write the constraints of the original LP 2. Create new variables called artificial variables 3. New objective: Minimize the sum of the artificials 4. Put the tableau in canonical form 5. Solve the Phase 1 Problem 6. Write the old objective function in the optimal tableau for the phase 1 problem. (delete artificials too) 7. Put the tableau in canonical form 8. Solve the original LP starting with the feasible bfs that you have obtained. 27

28 More on the phase 1 method What happens if w < 0 at the end of phase 1? It means the original problem is infeasible Is it possible that w = 0 at the end of phase 1 and there are still artificial variables in the basis? Yes, for technical reasons; but it is possible to find a new bfs with w = 0 and without the artificial variables. So, there is no real problem. The phase 1 method is widely used. There are bells and whistles to speed it up. An alternative approach: the big M method 28

29 The Big M-Method Starts off the same as the Phase 1 approach. It creates a new problem with artificial variables. x 1 x 2 x 3 x 4 x 5 x = = 2 29

30 The Big M-Method, continued Choose original objective function but penalize artificial variables maximize z = -3x 1 + 2x 2 + 4x 3 + x 4-99x 5 99x 6 z x 1 x 2 x 3 x 4 x 5 x = = = 2 If the profit is sufficiently negative, then an optimal solution will set x 5 = x 6 = 0. 30

31 Big M Method continued. Then one gets the basis into canonical form, and pivots to obtain an optimal solution to the new problem, which will also be optimal for the original problem. If some artificial variable is in the basis at the end, then make its profit even more negative and continue. If an artificial variable is positive no matter what we do, then there was no feasible solution to the original problem. 31

32 Is the Simplex Algorithm Finite? The number of basic feasible solutions is at most n! / (n-m)! m!, which is the number of ways of selecting m basic variables out of n. It is finite if we can guarantee that a sequence of pivots cannot lead back to the same bfs It is finite if we can guarantee that each bfs has a strictly better objective than the last bfs. The only thing preventing objective values from improving is degeneracy. 32

33 An improving pivot BV z x 1 x 2 x 3 x 4 z = 2 z = x 1 = 0 x = 6 x 2 = x 3 = 6-3 x = 2 x 4 = 2-2 Suppose that variable x 2 enters the basis. = 1. The objective value will increase by 2. If the RHS is strictly positive, the objective will improve. 33

34 A non-improving (degenerate) pivot BV z x 1 x 2 x 3 x 4 z = 2 z = x 1 = 0 x = 6 x 2 = x 3 = 6-3 x = 20 x 4 = 0-2 Suppose that variable x 2 enters the basis. = 0. The solution stays the same. But the basis will change. 34

35 Degeneracy BV z x 1 x 2 x 3 x 4 z = 2 x = 6 x = 20 A bfs is degenerate if b j = 0 for some j. Otherwise, it is non-degenerate. This bfs is degenerate. In a degenerate pivot, the solution may stay the same. 35

36 A degenerate pivot BV z x 1 x 2 x 3 x 4 z = 2 x /2 0 = 6 x /2 1 = 20 0 The entering variable is x 2. The exiting variable is the one in constraint 2. In this case the solution did not change. (But the tableau did.) In a non-degenerate pivot, z 0 increases. 36

37 Theorem. If every basic feasible solution is non-degenerate, the simplex method is finite. Fact. If bases can be degenerate, it is possible that the simplex method will cycle, that is repeat bases in a periodic manner. 37

38 Overview on how to make the simplex method finite. Reduce it to a previously solved problem Replace the LP by a similar LP that has the following features: The RHS of the new problem is nearly the same as the original problem. It differs by an almost infinitesimal amount every feasible basis for the perturbed problem is also a feasible basis for the original problem an optimal basis for the perturbed problem is also an optimal basis the original problem 38

39 Creating the new LP BV z x 1 x 2 x 3 x 4 z = 0 x = 6 +ε x = 2 +ε 2 In theory ε must be absurdly small (less than ) for this technique to work. In practice, the simplex algorithm doesn t repeat bases, and so they don t worry about implementing this approach. Perturbations will not be on the quiz, exam, or on homeworks. 39

40 z x 1 x 2 x 3 x 4 1 a = b = c 0 0 = = 5 To do with your partner (4 minutes) Give conditions on a, b and c so that the following are true. 1. The current bfs is the unique optimal solution. 2. The current bfs is optimal, but there are alternative optima. 3. The current bfs is degenerate. 4. The current bfs is non-optimal, and x 3 is pivoted out at the next iteration. 40

41 Some Remarks on Degeneracy and Alternative Optima It might seem that degeneracy would be rare. After all, why should we expect that the RHS would be 0 for some variable? In reality, degeneracy is incredibly common. The simplex method is not necessarily finite unless care is taken in the pivot rule. In reality, almost no one takes care, and the simplex method is not only finite, but it is incredibly efficient. The issue of alternative optima arises frequently, and is of importance in practice. 41

42 Overview The simplex method has been a huge success in optimization. It solves linear programs efficiently We can solve problems with millions of variables It can be a starting point for problems that are not linear The simplex method requires some simple techniques to get started Transformation into standard form Big M method or variant In practice it requires lots of engineering numerical stability choosing pivot rules that are fast in practice carrying out fast linear algebra CPLEX is one of several good LP codes 42

43 Summary of today s lecture The simplex method starts with an initial bfs and (under non-degeneracy) each pivot improves the objective function. It is very effective in practice. very good rules for choosing the entering variable very good implementations in practice to speed up the linear algebra Degeneracy: 0 s in RHS can lead to no improvement in the objective function. 0 s in the reduced costs can lead to alternative optima An artificial variable with a large negative cost can be used to create the first bfs. (Or use phase 1.) 43

44 Lecture check problems Section 4.6. Problems 1 and 2 Section 4.7. Problem 4 Section 4.8. Problem 1 Section Problem 2 Section Problem 4 44

Standard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta

Standard Form An LP is in standard form when: All variables are non-negativenegative All constraints are equalities Putting an LP formulation into sta Chapter 4 Linear Programming: The Simplex Method An Overview of the Simplex Method Standard Form Tableau Form Setting Up the Initial Simplex Tableau Improving the Solution Calculating the Next Tableau

More information

Review Solutions, Exam 2, Operations Research

Review Solutions, Exam 2, Operations Research Review Solutions, Exam 2, Operations Research 1. Prove the weak duality theorem: For any x feasible for the primal and y feasible for the dual, then... HINT: Consider the quantity y T Ax. SOLUTION: To

More information

IE 400: Principles of Engineering Management. Simplex Method Continued

IE 400: Principles of Engineering Management. Simplex Method Continued IE 400: Principles of Engineering Management Simplex Method Continued 1 Agenda Simplex for min problems Alternative optimal solutions Unboundedness Degeneracy Big M method Two phase method 2 Simplex for

More information

Dr. Maddah ENMG 500 Engineering Management I 10/21/07

Dr. Maddah ENMG 500 Engineering Management I 10/21/07 Dr. Maddah ENMG 500 Engineering Management I 10/21/07 Computational Procedure of the Simplex Method The optimal solution of a general LP problem is obtained in the following steps: Step 1. Express the

More information

Systems Analysis in Construction

Systems Analysis in Construction Systems Analysis in Construction CB312 Construction & Building Engineering Department- AASTMT by A h m e d E l h a k e e m & M o h a m e d S a i e d 3. Linear Programming Optimization Simplex Method 135

More information

Summary of the simplex method

Summary of the simplex method MVE165/MMG631,Linear and integer optimization with applications The simplex method: degeneracy; unbounded solutions; starting solutions; infeasibility; alternative optimal solutions Ann-Brith Strömberg

More information

February 22, Introduction to the Simplex Algorithm

February 22, Introduction to the Simplex Algorithm 15.53 February 22, 27 Introduction to the Simplex Algorithm 1 Quotes for today Give a man a fish and you feed him for a day. Teach him how to fish and you feed him for a lifetime. -- Lao Tzu Give a man

More information

AM 121: Intro to Optimization

AM 121: Intro to Optimization AM 121: Intro to Optimization Models and Methods Lecture 6: Phase I, degeneracy, smallest subscript rule. Yiling Chen SEAS Lesson Plan Phase 1 (initialization) Degeneracy and cycling Smallest subscript

More information

The Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006

The Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006 The Primal-Dual Algorithm P&S Chapter 5 Last Revised October 30, 2006 1 Simplex solves LP by starting at a Basic Feasible Solution (BFS) and moving from BFS to BFS, always improving the objective function,

More information

Special cases of linear programming

Special cases of linear programming Special cases of linear programming Infeasible solution Multiple solution (infinitely many solution) Unbounded solution Degenerated solution Notes on the Simplex tableau 1. The intersection of any basic

More information

1 Review Session. 1.1 Lecture 2

1 Review Session. 1.1 Lecture 2 1 Review Session Note: The following lists give an overview of the material that was covered in the lectures and sections. Your TF will go through these lists. If anything is unclear or you have questions

More information

Simplex Algorithm Using Canonical Tableaus

Simplex Algorithm Using Canonical Tableaus 41 Simplex Algorithm Using Canonical Tableaus Consider LP in standard form: Min z = cx + α subject to Ax = b where A m n has rank m and α is a constant In tableau form we record it as below Original Tableau

More information

AM 121: Intro to Optimization Models and Methods Fall 2018

AM 121: Intro to Optimization Models and Methods Fall 2018 AM 121: Intro to Optimization Models and Methods Fall 2018 Lecture 5: The Simplex Method Yiling Chen Harvard SEAS Lesson Plan This lecture: Moving towards an algorithm for solving LPs Tableau. Adjacent

More information

MATH 445/545 Homework 2: Due March 3rd, 2016

MATH 445/545 Homework 2: Due March 3rd, 2016 MATH 445/545 Homework 2: Due March 3rd, 216 Answer the following questions. Please include the question with the solution (write or type them out doing this will help you digest the problem). I do not

More information

The Simplex Method. Standard form (max) z c T x = 0 such that Ax = b.

The Simplex Method. Standard form (max) z c T x = 0 such that Ax = b. The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. The Simplex Method Standard form (max) z c T x = 0 such that Ax = b. Build initial tableau. z c T 0 0 A b The Simplex Method Standard

More information

Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras

Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Introduction to Operations Research Prof. G. Srinivasan Department of Management Studies Indian Institute of Technology, Madras Module - 03 Simplex Algorithm Lecture 15 Infeasibility In this class, we

More information

ORF 522. Linear Programming and Convex Analysis

ORF 522. Linear Programming and Convex Analysis ORF 5 Linear Programming and Convex Analysis Initial solution and particular cases Marco Cuturi Princeton ORF-5 Reminder: Tableaux At each iteration, a tableau for an LP in standard form keeps track of....................

More information

OPRE 6201 : 3. Special Cases

OPRE 6201 : 3. Special Cases OPRE 6201 : 3. Special Cases 1 Initialization: The Big-M Formulation Consider the linear program: Minimize 4x 1 +x 2 3x 1 +x 2 = 3 (1) 4x 1 +3x 2 6 (2) x 1 +2x 2 3 (3) x 1, x 2 0. Notice that there are

More information

Gauss-Jordan Elimination for Solving Linear Equations Example: 1. Solve the following equations: (3)

Gauss-Jordan Elimination for Solving Linear Equations Example: 1. Solve the following equations: (3) The Simple Method Gauss-Jordan Elimination for Solving Linear Equations Eample: Gauss-Jordan Elimination Solve the following equations: + + + + = 4 = = () () () - In the first step of the procedure, we

More information

Ann-Brith Strömberg. Lecture 4 Linear and Integer Optimization with Applications 1/10

Ann-Brith Strömberg. Lecture 4 Linear and Integer Optimization with Applications 1/10 MVE165/MMG631 Linear and Integer Optimization with Applications Lecture 4 Linear programming: degeneracy; unbounded solution; infeasibility; starting solutions Ann-Brith Strömberg 2017 03 28 Lecture 4

More information

3 The Simplex Method. 3.1 Basic Solutions

3 The Simplex Method. 3.1 Basic Solutions 3 The Simplex Method 3.1 Basic Solutions In the LP of Example 2.3, the optimal solution happened to lie at an extreme point of the feasible set. This was not a coincidence. Consider an LP in general form,

More information

Linear programs Optimization Geoff Gordon Ryan Tibshirani

Linear programs Optimization Geoff Gordon Ryan Tibshirani Linear programs 10-725 Optimization Geoff Gordon Ryan Tibshirani Review: LPs LPs: m constraints, n vars A: R m n b: R m c: R n x: R n ineq form [min or max] c T x s.t. Ax b m n std form [min or max] c

More information

The Simplex Method for Solving a Linear Program Prof. Stephen Graves

The Simplex Method for Solving a Linear Program Prof. Stephen Graves The Simplex Method for Solving a Linear Program Prof. Stephen Graves Observations from Geometry feasible region is a convex polyhedron an optimum occurs at a corner point possible algorithm - search over

More information

Variants of Simplex Method

Variants of Simplex Method Variants of Simplex Method All the examples we have used in the previous chapter to illustrate simple algorithm have the following common form of constraints; i.e. a i x + a i x + + a in x n b i, i =,,,m

More information

min 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14

min 4x 1 5x 2 + 3x 3 s.t. x 1 + 2x 2 + x 3 = 10 x 1 x 2 6 x 1 + 3x 2 + x 3 14 The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. If necessary,

More information

Prelude to the Simplex Algorithm. The Algebraic Approach The search for extreme point solutions.

Prelude to the Simplex Algorithm. The Algebraic Approach The search for extreme point solutions. Prelude to the Simplex Algorithm The Algebraic Approach The search for extreme point solutions. 1 Linear Programming-1 x 2 12 8 (4,8) Max z = 6x 1 + 4x 2 Subj. to: x 1 + x 2

More information

Lecture slides by Kevin Wayne

Lecture slides by Kevin Wayne LINEAR PROGRAMMING I a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm Lecture slides by Kevin Wayne Last updated on 7/25/17 11:09 AM Linear programming

More information

Simplex method(s) for solving LPs in standard form

Simplex method(s) for solving LPs in standard form Simplex method: outline I The Simplex Method is a family of algorithms for solving LPs in standard form (and their duals) I Goal: identify an optimal basis, as in Definition 3.3 I Versions we will consider:

More information

Lecture 5 Simplex Method. September 2, 2009

Lecture 5 Simplex Method. September 2, 2009 Simplex Method September 2, 2009 Outline: Lecture 5 Re-cap blind search Simplex method in steps Simplex tableau Operations Research Methods 1 Determining an optimal solution by exhaustive search Lecture

More information

21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20.

21. Solve the LP given in Exercise 19 using the big-m method discussed in Exercise 20. Extra Problems for Chapter 3. Linear Programming Methods 20. (Big-M Method) An alternative to the two-phase method of finding an initial basic feasible solution by minimizing the sum of the artificial

More information

Simplex Method for LP (II)

Simplex Method for LP (II) Simplex Method for LP (II) Xiaoxi Li Wuhan University Sept. 27, 2017 (week 4) Operations Research (Li, X.) Simplex Method for LP (II) Sept. 27, 2017 (week 4) 1 / 31 Organization of this lecture Contents:

More information

Linear Programming and the Simplex method

Linear Programming and the Simplex method Linear Programming and the Simplex method Harald Enzinger, Michael Rath Signal Processing and Speech Communication Laboratory Jan 9, 2012 Harald Enzinger, Michael Rath Jan 9, 2012 page 1/37 Outline Introduction

More information

Lecture 11: Post-Optimal Analysis. September 23, 2009

Lecture 11: Post-Optimal Analysis. September 23, 2009 Lecture : Post-Optimal Analysis September 23, 2009 Today Lecture Dual-Simplex Algorithm Post-Optimal Analysis Chapters 4.4 and 4.5. IE 30/GE 330 Lecture Dual Simplex Method The dual simplex method will

More information

LINEAR PROGRAMMING I. a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm

LINEAR PROGRAMMING I. a refreshing example standard form fundamental questions geometry linear algebra simplex algorithm Linear programming Linear programming. Optimize a linear function subject to linear inequalities. (P) max c j x j n j= n s. t. a ij x j = b i i m j= x j 0 j n (P) max c T x s. t. Ax = b Lecture slides

More information

CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination

CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 22th June 2005 Chapter 8: Finite Termination Recap On Monday, we established In the absence of degeneracy, the simplex method will

More information

Linear Programming. Linear Programming I. Lecture 1. Linear Programming. Linear Programming

Linear Programming. Linear Programming I. Lecture 1. Linear Programming. Linear Programming Linear Programming Linear Programming Lecture Linear programming. Optimize a linear function subject to linear inequalities. (P) max " c j x j n j= n s. t. " a ij x j = b i # i # m j= x j 0 # j # n (P)

More information

TIM 206 Lecture 3: The Simplex Method

TIM 206 Lecture 3: The Simplex Method TIM 206 Lecture 3: The Simplex Method Kevin Ross. Scribe: Shane Brennan (2006) September 29, 2011 1 Basic Feasible Solutions Have equation Ax = b contain more columns (variables) than rows (constraints),

More information

CO 602/CM 740: Fundamentals of Optimization Problem Set 4

CO 602/CM 740: Fundamentals of Optimization Problem Set 4 CO 602/CM 740: Fundamentals of Optimization Problem Set 4 H. Wolkowicz Fall 2014. Handed out: Wednesday 2014-Oct-15. Due: Wednesday 2014-Oct-22 in class before lecture starts. Contents 1 Unique Optimum

More information

The Simplex Method. Lecture 5 Standard and Canonical Forms and Setting up the Tableau. Lecture 5 Slide 1. FOMGT 353 Introduction to Management Science

The Simplex Method. Lecture 5 Standard and Canonical Forms and Setting up the Tableau. Lecture 5 Slide 1. FOMGT 353 Introduction to Management Science The Simplex Method Lecture 5 Standard and Canonical Forms and Setting up the Tableau Lecture 5 Slide 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard

More information

Math 273a: Optimization The Simplex method

Math 273a: Optimization The Simplex method Math 273a: Optimization The Simplex method Instructor: Wotao Yin Department of Mathematics, UCLA Fall 2015 material taken from the textbook Chong-Zak, 4th Ed. Overview: idea and approach If a standard-form

More information

Slack Variable. Max Z= 3x 1 + 4x 2 + 5X 3. Subject to: X 1 + X 2 + X x 1 + 4x 2 + X X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0

Slack Variable. Max Z= 3x 1 + 4x 2 + 5X 3. Subject to: X 1 + X 2 + X x 1 + 4x 2 + X X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0 Simplex Method Slack Variable Max Z= 3x 1 + 4x 2 + 5X 3 Subject to: X 1 + X 2 + X 3 20 3x 1 + 4x 2 + X 3 15 2X 1 + X 2 + 4X 3 10 X 1 0, X 2 0, X 3 0 Standard Form Max Z= 3x 1 +4x 2 +5X 3 + 0S 1 + 0S 2

More information

Distributed Real-Time Control Systems. Lecture Distributed Control Linear Programming

Distributed Real-Time Control Systems. Lecture Distributed Control Linear Programming Distributed Real-Time Control Systems Lecture 13-14 Distributed Control Linear Programming 1 Linear Programs Optimize a linear function subject to a set of linear (affine) constraints. Many problems can

More information

MATH2070 Optimisation

MATH2070 Optimisation MATH2070 Optimisation Linear Programming Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The standard Linear Programming (LP) Problem Graphical method of solving LP problem

More information

ORF 307: Lecture 2. Linear Programming: Chapter 2 Simplex Methods

ORF 307: Lecture 2. Linear Programming: Chapter 2 Simplex Methods ORF 307: Lecture 2 Linear Programming: Chapter 2 Simplex Methods Robert Vanderbei February 8, 2018 Slides last edited on February 8, 2018 http://www.princeton.edu/ rvdb Simplex Method for LP An Example.

More information

IE 5531: Engineering Optimization I

IE 5531: Engineering Optimization I IE 5531: Engineering Optimization I Lecture 5: The Simplex method, continued Prof. John Gunnar Carlsson September 22, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 22, 2010

More information

Metode Kuantitatif Bisnis. Week 4 Linear Programming Simplex Method - Minimize

Metode Kuantitatif Bisnis. Week 4 Linear Programming Simplex Method - Minimize Metode Kuantitatif Bisnis Week 4 Linear Programming Simplex Method - Minimize Outlines Solve Linear Programming Model Using Graphic Solution Solve Linear Programming Model Using Simplex Method (Maximize)

More information

Farkas Lemma, Dual Simplex and Sensitivity Analysis

Farkas Lemma, Dual Simplex and Sensitivity Analysis Summer 2011 Optimization I Lecture 10 Farkas Lemma, Dual Simplex and Sensitivity Analysis 1 Farkas Lemma Theorem 1. Let A R m n, b R m. Then exactly one of the following two alternatives is true: (i) x

More information

Dual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP:

Dual Basic Solutions. Observation 5.7. Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Dual Basic Solutions Consider LP in standard form with A 2 R m n,rank(a) =m, and dual LP: Observation 5.7. AbasisB yields min c T x max p T b s.t. A x = b s.t. p T A apple c T x 0 aprimalbasicsolutiongivenbyx

More information

Week_4: simplex method II

Week_4: simplex method II Week_4: simplex method II 1 1.introduction LPs in which all the constraints are ( ) with nonnegative right-hand sides offer a convenient all-slack starting basic feasible solution. Models involving (=)

More information

MATH 445/545 Test 1 Spring 2016

MATH 445/545 Test 1 Spring 2016 MATH 445/545 Test Spring 06 Note the problems are separated into two sections a set for all students and an additional set for those taking the course at the 545 level. Please read and follow all of these

More information

March 13, Duality 3

March 13, Duality 3 15.53 March 13, 27 Duality 3 There are concepts much more difficult to grasp than duality in linear programming. -- Jim Orlin The concept [of nonduality], often described in English as "nondualism," is

More information

Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method

Yinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapters 2.3-2.5, 3.1-3.4) 1 Geometry of Linear

More information

Ω R n is called the constraint set or feasible set. x 1

Ω R n is called the constraint set or feasible set. x 1 1 Chapter 5 Linear Programming (LP) General constrained optimization problem: minimize subject to f(x) x Ω Ω R n is called the constraint set or feasible set. any point x Ω is called a feasible point We

More information

CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination

CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination CO350 Linear Programming Chapter 8: Degeneracy and Finite Termination 27th June 2005 Chapter 8: Finite Termination 1 The perturbation method Recap max c T x (P ) s.t. Ax = b x 0 Assumption: B is a feasible

More information

1 Overview. 2 Extreme Points. AM 221: Advanced Optimization Spring 2016

1 Overview. 2 Extreme Points. AM 221: Advanced Optimization Spring 2016 AM 22: Advanced Optimization Spring 206 Prof. Yaron Singer Lecture 7 February 7th Overview In the previous lectures we saw applications of duality to game theory and later to learning theory. In this lecture

More information

Lecture 4: Algebra, Geometry, and Complexity of the Simplex Method. Reading: Sections 2.6.4, 3.5,

Lecture 4: Algebra, Geometry, and Complexity of the Simplex Method. Reading: Sections 2.6.4, 3.5, Lecture 4: Algebra, Geometry, and Complexity of the Simplex Method Reading: Sections 2.6.4, 3.5, 10.2 10.5 1 Summary of the Phase I/Phase II Simplex Method We write a typical simplex tableau as z x 1 x

More information

(P ) Minimize 4x 1 + 6x 2 + 5x 3 s.t. 2x 1 3x 3 3 3x 2 2x 3 6

(P ) Minimize 4x 1 + 6x 2 + 5x 3 s.t. 2x 1 3x 3 3 3x 2 2x 3 6 The exam is three hours long and consists of 4 exercises. The exam is graded on a scale 0-25 points, and the points assigned to each question are indicated in parenthesis within the text. Problem 1 Consider

More information

Lecture 2: The Simplex method

Lecture 2: The Simplex method Lecture 2 1 Linear and Combinatorial Optimization Lecture 2: The Simplex method Basic solution. The Simplex method (standardform, b>0). 1. Repetition of basic solution. 2. One step in the Simplex algorithm.

More information

56:171 Operations Research Fall 1998

56:171 Operations Research Fall 1998 56:171 Operations Research Fall 1998 Quiz Solutions D.L.Bricker Dept of Mechanical & Industrial Engineering University of Iowa 56:171 Operations Research Quiz

More information

OPERATIONS RESEARCH. Linear Programming Problem

OPERATIONS RESEARCH. Linear Programming Problem OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com MODULE - 2: Simplex Method for

More information

Chapter 4 The Simplex Algorithm Part II

Chapter 4 The Simplex Algorithm Part II Chapter 4 The Simple Algorithm Part II Based on Introduction to Mathematical Programming: Operations Research, Volume 4th edition, by Wayne L Winston and Munirpallam Venkataramanan Lewis Ntaimo L Ntaimo

More information

Part III: A Simplex pivot

Part III: A Simplex pivot MA 3280 Lecture 31 - More on The Simplex Method Friday, April 25, 2014. Objectives: Analyze Simplex examples. We were working on the Simplex tableau The matrix form of this system of equations is called

More information

Part 1. The Review of Linear Programming

Part 1. The Review of Linear Programming In the name of God Part 1. The Review of Linear Programming 1.2. Spring 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Basic Feasible Solutions Key to the Algebra of the The Simplex Algorithm

More information

Summary of the simplex method

Summary of the simplex method MVE165/MMG630, The simplex method; degeneracy; unbounded solutions; infeasibility; starting solutions; duality; interpretation Ann-Brith Strömberg 2012 03 16 Summary of the simplex method Optimality condition:

More information

In Chapters 3 and 4 we introduced linear programming

In Chapters 3 and 4 we introduced linear programming SUPPLEMENT The Simplex Method CD3 In Chapters 3 and 4 we introduced linear programming and showed how models with two variables can be solved graphically. We relied on computer programs (WINQSB, Excel,

More information

ECE 307 Techniques for Engineering Decisions

ECE 307 Techniques for Engineering Decisions ECE 7 Techniques for Engineering Decisions Introduction to the Simple Algorithm George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign ECE 7 5 9 George

More information

Introduce the idea of a nondegenerate tableau and its analogy with nondenegerate vertices.

Introduce the idea of a nondegenerate tableau and its analogy with nondenegerate vertices. 2 JORDAN EXCHANGE REVIEW 1 Lecture Outline The following lecture covers Section 3.5 of the textbook [?] Review a labeled Jordan exchange with pivoting. Introduce the idea of a nondegenerate tableau and

More information

1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations

1. Algebraic and geometric treatments Consider an LP problem in the standard form. x 0. Solutions to the system of linear equations The Simplex Method Most textbooks in mathematical optimization, especially linear programming, deal with the simplex method. In this note we study the simplex method. It requires basically elementary linear

More information

3. THE SIMPLEX ALGORITHM

3. THE SIMPLEX ALGORITHM Optimization. THE SIMPLEX ALGORITHM DPK Easter Term. Introduction We know that, if a linear programming problem has a finite optimal solution, it has an optimal solution at a basic feasible solution (b.f.s.).

More information

Math 354 Summer 2004 Solutions to review problems for Midterm #1

Math 354 Summer 2004 Solutions to review problems for Midterm #1 Solutions to review problems for Midterm #1 First: Midterm #1 covers Chapter 1 and 2. In particular, this means that it does not explicitly cover linear algebra. Also, I promise there will not be any proofs.

More information

4.5 Simplex method. LP in standard form: min z = c T x s.t. Ax = b

4.5 Simplex method. LP in standard form: min z = c T x s.t. Ax = b 4.5 Simplex method LP in standard form: min z = c T x s.t. Ax = b x 0 George Dantzig (1914-2005) Examine a sequence of basic feasible solutions with non increasing objective function values until an optimal

More information

Lecture 5. x 1,x 2,x 3 0 (1)

Lecture 5. x 1,x 2,x 3 0 (1) Computational Intractability Revised 2011/6/6 Lecture 5 Professor: David Avis Scribe:Ma Jiangbo, Atsuki Nagao 1 Duality The purpose of this lecture is to introduce duality, which is an important concept

More information

Optimization (168) Lecture 7-8-9

Optimization (168) Lecture 7-8-9 Optimization (168) Lecture 7-8-9 Jesús De Loera UC Davis, Mathematics Wednesday, April 2, 2012 1 DEGENERACY IN THE SIMPLEX METHOD 2 DEGENERACY z =2x 1 x 2 + 8x 3 x 4 =1 2x 3 x 5 =3 2x 1 + 4x 2 6x 3 x 6

More information

2.098/6.255/ Optimization Methods Practice True/False Questions

2.098/6.255/ Optimization Methods Practice True/False Questions 2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 1-3 line supporting sentence

More information

9.1 Linear Programs in canonical form

9.1 Linear Programs in canonical form 9.1 Linear Programs in canonical form LP in standard form: max (LP) s.t. where b i R, i = 1,..., m z = j c jx j j a ijx j b i i = 1,..., m x j 0 j = 1,..., n But the Simplex method works only on systems

More information

MATH 4211/6211 Optimization Linear Programming

MATH 4211/6211 Optimization Linear Programming MATH 4211/6211 Optimization Linear Programming Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 The standard form of a Linear

More information

The Simplex Algorithm and Goal Programming

The Simplex Algorithm and Goal Programming The Simplex Algorithm and Goal Programming In Chapter 3, we saw how to solve two-variable linear programming problems graphically. Unfortunately, most real-life LPs have many variables, so a method is

More information

Chap6 Duality Theory and Sensitivity Analysis

Chap6 Duality Theory and Sensitivity Analysis Chap6 Duality Theory and Sensitivity Analysis The rationale of duality theory Max 4x 1 + x 2 + 5x 3 + 3x 4 S.T. x 1 x 2 x 3 + 3x 4 1 5x 1 + x 2 + 3x 3 + 8x 4 55 x 1 + 2x 2 + 3x 3 5x 4 3 x 1 ~x 4 0 If we

More information

Chapter 4 The Simplex Algorithm Part I

Chapter 4 The Simplex Algorithm Part I Chapter 4 The Simplex Algorithm Part I Based on Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan Lewis Ntaimo 1 Modeling

More information

AM 121: Intro to Optimization Models and Methods

AM 121: Intro to Optimization Models and Methods AM 121: Intro to Optimization Models and Methods Fall 2017 Lecture 2: Intro to LP, Linear algebra review. Yiling Chen SEAS Lecture 2: Lesson Plan What is an LP? Graphical and algebraic correspondence Problems

More information

Math Models of OR: Some Definitions

Math Models of OR: Some Definitions Math Models of OR: Some Definitions John E. Mitchell Department of Mathematical Sciences RPI, Troy, NY 12180 USA September 2018 Mitchell Some Definitions 1 / 20 Active constraints Outline 1 Active constraints

More information

The Simplex Algorithm: Technicalities 1

The Simplex Algorithm: Technicalities 1 1/45 The Simplex Algorithm: Technicalities 1 Adrian Vetta 1 This presentation is based upon the book Linear Programming by Vasek Chvatal 2/45 Two Issues Here we discuss two potential problems with the

More information

A Review of Linear Programming

A Review of Linear Programming A Review of Linear Programming Instructor: Farid Alizadeh IEOR 4600y Spring 2001 February 14, 2001 1 Overview In this note we review the basic properties of linear programming including the primal simplex

More information

Termination, Cycling, and Degeneracy

Termination, Cycling, and Degeneracy Chapter 4 Termination, Cycling, and Degeneracy We now deal first with the question, whether the simplex method terminates. The quick answer is no, if it is implemented in a careless way. Notice that we

More information

Dr. S. Bourazza Math-473 Jazan University Department of Mathematics

Dr. S. Bourazza Math-473 Jazan University Department of Mathematics Dr. Said Bourazza Department of Mathematics Jazan University 1 P a g e Contents: Chapter 0: Modelization 3 Chapter1: Graphical Methods 7 Chapter2: Simplex method 13 Chapter3: Duality 36 Chapter4: Transportation

More information

Linear Programming The Simplex Algorithm: Part II Chapter 5

Linear Programming The Simplex Algorithm: Part II Chapter 5 1 Linear Programming The Simplex Algorithm: Part II Chapter 5 University of Chicago Booth School of Business Kipp Martin October 17, 2017 Outline List of Files Key Concepts Revised Simplex Revised Simplex

More information

Understanding the Simplex algorithm. Standard Optimization Problems.

Understanding the Simplex algorithm. Standard Optimization Problems. Understanding the Simplex algorithm. Ma 162 Spring 2011 Ma 162 Spring 2011 February 28, 2011 Standard Optimization Problems. A standard maximization problem can be conveniently described in matrix form

More information

3. Duality: What is duality? Why does it matter? Sensitivity through duality.

3. Duality: What is duality? Why does it matter? Sensitivity through duality. 1 Overview of lecture (10/5/10) 1. Review Simplex Method 2. Sensitivity Analysis: How does solution change as parameters change? How much is the optimal solution effected by changing A, b, or c? How much

More information

Example. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive

Example. 1 Rows 1,..., m of the simplex tableau remain lexicographically positive 3.4 Anticycling Lexicographic order In this section we discuss two pivoting rules that are guaranteed to avoid cycling. These are the lexicographic rule and Bland s rule. Definition A vector u R n is lexicographically

More information

Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004

Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004 Linear Programming Duality P&S Chapter 3 Last Revised Nov 1, 2004 1 In this section we lean about duality, which is another way to approach linear programming. In particular, we will see: How to define

More information

Non-Standard Constraints. Setting up Phase 1 Phase 2

Non-Standard Constraints. Setting up Phase 1 Phase 2 Non-Standard Constraints Setting up Phase 1 Phase 2 Maximizing with Mixed Constraints Some maximization problems contain mixed constraints, like this: maximize 3x 1 + 2x 2 subject to 2x 1 + x 2 50 (standard)

More information

Simplex Method: More Details

Simplex Method: More Details Chapter 4 Simplex Method: More Details 4. How to Start Simplex? Two Phases Our simplex algorithm is constructed for solving the following standard form of linear programs: min c T x s.t. Ax = b x 0 where

More information

The simplex algorithm

The simplex algorithm The simplex algorithm The simplex algorithm is the classical method for solving linear programs. Its running time is not polynomial in the worst case. It does yield insight into linear programs, however,

More information

Chapter 3, Operations Research (OR)

Chapter 3, Operations Research (OR) Chapter 3, Operations Research (OR) Kent Andersen February 7, 2007 1 Linear Programs (continued) In the last chapter, we introduced the general form of a linear program, which we denote (P) Minimize Z

More information

Lecture 14: October 11

Lecture 14: October 11 10-725: Optimization Fall 2012 Lecture 14: October 11 Lecturer: Geoff Gordon/Ryan Tibshirani Scribes: Zitao Liu Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not

More information

Notes on Simplex Algorithm

Notes on Simplex Algorithm Notes on Simplex Algorithm CS 9 Staff October 8, 7 Until now, we have represented the problems geometrically, and solved by finding a corner and moving around Now we learn an algorithm to solve this without

More information

ORF 522. Linear Programming and Convex Analysis

ORF 522. Linear Programming and Convex Analysis ORF 522 Linear Programming and Convex Analysis The Simplex Method Marco Cuturi Princeton ORF-522 1 Reminder: Basic Feasible Solutions, Extreme points, Optima Some important theorems last time for standard

More information

15-780: LinearProgramming

15-780: LinearProgramming 15-780: LinearProgramming J. Zico Kolter February 1-3, 2016 1 Outline Introduction Some linear algebra review Linear programming Simplex algorithm Duality and dual simplex 2 Outline Introduction Some linear

More information

Contents. 4.5 The(Primal)SimplexMethod NumericalExamplesoftheSimplexMethod

Contents. 4.5 The(Primal)SimplexMethod NumericalExamplesoftheSimplexMethod Contents 4 The Simplex Method for Solving LPs 149 4.1 Transformations to be Carried Out On an LP Model Before Applying the Simplex Method On It... 151 4.2 Definitions of Various Types of Basic Vectors

More information

The Simplex Method. Formulate Constrained Maximization or Minimization Problem. Convert to Standard Form. Convert to Canonical Form

The Simplex Method. Formulate Constrained Maximization or Minimization Problem. Convert to Standard Form. Convert to Canonical Form The Simplex Method 1 The Simplex Method Formulate Constrained Maximization or Minimization Problem Convert to Standard Form Convert to Canonical Form Set Up the Tableau and the Initial Basic Feasible Solution

More information